The phrase “wave based measurement” refers to the measurement of the nature of a wave at a receiver from a wave created by a source. Wave based measurements have many applications, some of which are as simple as measuring the distance from a source to a receiver. Such simple measurements are based on a measurement of the time the wave travels and, knowing the speed at which the wave travels (i.e., propagates), the distance is computed as the product of the known wave speed and the elapsed travel time.
Simple wave-based measurements, such as the one mentioned above, require that the wave travel through a homogeneous medium, i.e., that the material through which the wave travels have a uniform wave speed. When a wave travels through a medium where the wave speed (or other attributes) is not constant, the character of the recorded wave is altered. While this complication makes it impossible to use simple wave-based measurements to determine distant, it offers the possibility of applying more sophisticated analyses to characterize the medium. Such techniques are in regular use for diagnostic medicine, non-destructive evaluation (NDE), and geophysics.
What makes these applications so powerful is that changes in wave propagation characteristics are diagnostics for more fundamental material properties. For example, ultrasound has many uses in diagnostic medicine and one such use is the detection of tumors in the breast. Tumors have a density that is typically greater than healthy soft tissue, and this density difference change the propagation of high frequency acoustic waves. Similarly, differences in material properties are exploited in wave-based NDE to identify impurities or micro-fractures, and are used in geophysics to identify buried man-made objects or geologic structures such as minerals, oil, or natural gas deposits.
In most wave-based applications, the full extent of the information that is encoded in the data is not recovered. Typically, there are several material properties that alter the propagation of waves and, when considered individually, these properties serve to more completely characterize objects such as tumors. With electromagnetic waves, changes in wave character result from changes in both dielectric and electrical conductivity. Having access to each of these properties separately, rather than as a composite response, better serves to characterize the material of interest, such as distinguishing plastic from metal and wood, etc. Attenuation is a property that causes a loss of wave energy and, with electromagnetic waves, electrical conductivity causes wave attenuation. In low frequency acoustics (seismics), attenuation can be indicative of the presence of certain types of materials, most notably hydrocarbons, so that the capacity to isolate the contribution of attenuation in wave propagation can be considered a direct indicator of hydrocarbons.
It is currently possible to separate fundamental material properties from wave-based measurements. However, this capability is limited to very specific measurement configurations where there is a certain type of measurement symmetry. One example is where arrays of both sources and receivers are distributed around the circumference of a ring. A ray path is defined to be a direction of wave propagation from a source to receiver. When considering a source on one side of the ring and a receiver on the opposite side, it is clear that, within this array geometry, other sources and receivers can be paired so as to measure ray paths both opposite and perpendicular to the path of interest. A second example of symmetric geometry is an array of sources positioned along one line and an array of receivers distributed over a parallel line some distance away. Sources and receivers can be paired such that for every ray path taken to be downward from left to right, a similar ray path can be captured that is upward from left to right.
There are many other measurement configurations where this symmetry does not exist and for which there has been no way of separately extracting fundamental material properties. (Such configurations are referred to as limited view configurations.) Returning to the example regarding the detection of breast tumors discussed above, while it may be possible to completely surround some portion of the breast with a ring of ultrasonic transducers, there are portions of the breast as well as other areas of the body, such as heart, liver, kidneys, etc., that cannot be non-invasively accessed in this manner and, thus, measurements must be with limited views. Another limited view geometry is reflection where both sources and receivers are typically distributed over the same or adjacent lines. Reflection geometries are common in medical ultrasound (obstetrics, for example) and many geophysical applications such ground penetrating radar and seismic reflection (a mainstay of resource exploration).
What is needed is a methodology that will allow the isolation of individual material properties from limited view measurements. Such a methodology will have broad applications in many types of wave-based measurements.